Definition:Correlation Coefficient

Definition

A correlation coefficient is a statistical coefficient which provides a measure of a particular aspect of correlation amongst a set of data.

Pearson Correlation Coefficient

Also known as the product-moment correlation coefficient.

Let $X$ and $Y$ be random variables.

Let the variances of $X$ and $Y$ exist and be finite.

Then the Pearson correlation coefficient of $X$ and $Y$, typically denoted $\map \rho {X, Y}$, is defined by:

$\map \rho {X, Y} = \dfrac {\cov {X, Y} } {\sqrt {\var X \, \var Y} }$

where $\cov {X, Y}$ is the covariance of $X$ and $Y$.

Spearman's Rank Correlation Coefficient

Let $X$ and $Y$ be two rankings assigned to the same set of entities.

The Spearman's rank correlation coefficient is the Pearson correlation coefficient between $X$ and $Y$.

Kendall's Rank Correlation Coefficient

Kendall's rank correlation coefficient is a test for consistency of $2$ sets of rankings $\sequence a_n$ and $\sequence b_n$ on a set $S$ of $n$ objects.

The set $R$ of ordered pairs $\tuple {a_i, b_i}$ is assembled:

$R = \set {\tuple {a_i, b_i}: i \in \set {1, 2, \ldots, n} }$

and ordered according to $\sequence a$.

The number $Q$ of elements of $S$ out of ranking order from $\sequence b$ is counted.

Kendall's rank correlation coefficient is then formed:

$K = 1 - \dfrac {4 Q} {n \paren {n + 1} }$

which takes values between $-1$ (complete disagreement) and $+1$ (complete agreement).

Complete disagreement happens when $\sequence a_n$ is in reverse order to $\sequence b_n$.

Biserial Correlation Coefficient

A biserial correlation coefficient is a measure of dependence between:

a continuous random variable $X$

a discrete random variable $Y$ which can take one of only $2$ values: $y_1$ and $y_2$.

Also known as

A correlation coefficient is also known as a cofficient of correlation.

Also see

• Results about correlation coefficients can be found here.